#371414

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,4,3)(13,14,18,21,17,19,16,25)(15,22,24,20,23,28,26,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(16,18)(22,28);(6,9)(10,12)(13,15)(14,20)(16,26)(17,23)(18,24)(19,27)(21,28)(22,25);(5,7,8,11)(6,10,9,12)(14,19)(16,18)(20,27)(24,26)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278610);